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Stream: learning: questions

Topic: Categorification of eigenvalues and eigenvectors

fosco (Apr 20 2025 at 15:30):

I want to submit to you the following shenanigan which I find very funny to play with.

Let $p : Y^o\times Y\to Set$ be an endoprofunctor on a category $Y$. Let $\Lambda$ a set, and denote still as $\Lambda$ the costant functor at $\Lambda$.

• $\Lambda$ is an eigenvalue of $p$ if the category $(p/\Lambda)$ of natural transformations $p \Rightarrow \Lambda$ is nonempty and... something else I'd like to figure out so that what follows makes sense (and avoids vacuous truths of sort).
• A functor $F : Y^{op}\to Set$ is an eigenvector wrt $\Lambda$ if, when $\Lambda$ is an eigenvalue, for every $p\Rightarrow\Lambda$ in $(p/\Lambda)$, there is an initial cowedge $p(x,y)\times Fy \to \Lambda\times Fx$ (so that the "matrix product" $(p\otimes F)x:= \int^y p(x,y)\times Fy$ between the matrix $p$ and the vector $F$ is canonically isomorphic to the product $\Lambda\times F$ of the "scalar" $\Lambda$ and of $F$, evaluated at $x$, as prescribed by the theory of eigenvalues/eigenvectors).

fosco (Apr 20 2025 at 15:33):

I have a hunch that connectedness, or some sort of contractibility, of $(p/\Lambda)$ could be required; I also suspect the second property is too strong, but note that (for example) with this definition the eigenspace of $\Lambda$ for $p$ (the subcategory spanned by all eigenvectors wrt $\Lambda$) is closed under colimits, so it can't be too trivial a subcategory of the small-cocomplete category $[Y^{op},Set]$. Note also that the only eigenvector of hom is the terminal object, with eigenspace the whole category, as god intended.

fosco (Apr 20 2025 at 15:35):

I remember a few people that, like me, love categorifying linear algebra; @Simon Willerton might have tinkered with this idea, or maybe @Jade Master (who I remember was blogging about the analogies between matrices and quantale-enriched profunctors?)